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European Journal of Mechanics A/Solids 28 (2009) 14–24

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European Journal of Mechanics A/Solids

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The dynamic response of clamped rectangular Y-frame and corrugated coresandwich plates

V. Rubino, V.S. Deshpande ∗, N.A. Fleck

Cambridge University, Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 March 2008Accepted 7 June 2008Available online 3 July 2008

Keywords:Sandwich platesDynamic responseFE simulationsLattice materials

The dynamic response of fully clamped, monolithic and sandwich plates of equal areal mass has beenmeasured by loading rectangular plates over a central patch with metal foam projectiles. All platesare made from AISI 304 stainless steel, and the sandwich topologies comprise two identical face-sheetsand either Y-frame or corrugated cores. The resistance to shock loading is quantified by the permanenttransverse deflection at mid-span of the plates as a function of projectile momentum. At low levels ofprojectile momentum both types of sandwich plate deflect less than monolithic plates of equal arealmass. However, at higher levels of projectile momentum, the sandwich plates tear while the monolithicplates remain intact. Three-dimensional finite element (FE) calculations adequately predict the measuredresponses, prior to the onset of tearing. These calculations also reveal that the accumulated plastic strainsin the front face of the sandwich plates exceed those in the monolithic plates. These high plastic strainslead to failure of the front face sheets of the sandwich plates at lower values of projectile momentumthan for the equivalent monolithic plates.

© 2008 Elsevier Masson SAS. All rights reserved.

1. Introduction

Clamped plates are representative of the structures used in thedesign of commercial and military vehicles. For example, the hullof a ship comprises plates welded to an array of stiffeners. A de-tailed overview on the shock response of monolithic beams andplates has been given by Jones (1989). More particularly Wangand Hopkins (1954) and Symmonds (1954) have analysed the im-pulsive response of clamped circular plates and beams. However,their analyses were restricted to small deflections and linear bend-ing kinematics. By direct application of the principle of virtualwork for an assumed deformation mode, Jones (1971) presentedan approximate solution for simply supported circular monolithicplates undergoing finite deflections. Experiments on impulsivelyloaded beams and plates have confirmed the deformation modespredicted by these analyses and also revealed a range of failuremodes (Menkes and Opat, 1973; Nurick and Shave, 2000). Theoret-ical studies by Lee and Wierzbicki (2005a, 2005b) have analysedthe so-called discing and petalling failure modes in impulsivelyloaded clamped plates while Balden and Nurick (2005) have anal-ysed the so-called shear rupture modes.

Over the past decade there have been substantial changes inship design with double hulls and sandwich hulls replacing mono-lithic plates, see for example the review by Paik (2003). One such

* Corresponding author. Fax: +44 1223 332662.E-mail address: [email protected] (V.S. Deshpande).

0997-7538/$ – see front matter © 2008 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechsol.2008.06.001

innovation has been the Schelde Shipbuilding1 sandwich design ofship hulls with a Y-frame core, see Fig. 1a. Full-scale ship collisiontrials reveal that the Y-frame design is more resistant to tearingthan conventional monolithic designs, see Wevers and Vredeveldt(1999) and Ludolphy (2001). Likewise, the finite element simu-lations by Konter et al. (2004) suggest that the Y-frame confersimproved perforation resistance. Naar et al. (2002) have arguedin broad terms that the ability of the bending-governed Y-frametopology to spread the impact load over a wide area, combinedwith the high in-plane stretching resistance of the Y-frame, givesthe enhanced performance of the Y-frame sandwich constructionover conventional single and double hull designs. The corrugatedsandwich core (Fig. 1b) is a competing prismatic topology to thatof the Y-frame.

Pedersen et al. (2006) and Rubino et al. (2008a) have inves-tigated the quasi-static structural response of Y-frame sandwichbeams while Côté et al. (2006) studied the response of sandwichbeams with a corrugated core. These studies reveal that geometri-cal imperfections and/or non-uniform loading (such as indentationloading) induce bending within the struts of the corrugated coreand reduce the compressive strength of the corrugated core to ap-proximately that of the Y-frame core. Tilbrook et al. (2007) haveperformed a combined experimental and numerical investigationinto the dynamic compressive response of these cores. They foundthat buckling of the webs is delayed at low impact velocities by

1 Royal Schelde, P.O. Box 16 4380 AA Vlissingen, The Netherlands.

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V. Rubino et al. / European Journal of Mechanics A/Solids 28 (2009) 14–24 15

(a) (b)

Fig. 1. Sketch of the (a) Y-frame and (b) corrugated sandwich cores as used in ship hull construction. The core is sandwiched between the outer and inner hull of the ship.

(a)

(b)

(c)

Fig. 2. (a) Sketch of the clamped sandwich plate geometry and the loading arrangement. Geometry of half sandwich plate for (b) Y-frame and (c) corrugated sandwich coresinvestigated in this study. The cross-section is also shown for each topology. All dimensions are in mm.

inertial stabilisation of the webs while plastic shock wave effectsdominate the response at higher impact velocities.

Radford et al. (2005) have developed an experimental techniqueto subject structures to high intensity pressure pulses using metal

16 V. Rubino et al. / European Journal of Mechanics A/Solids 28 (2009) 14–24

(a) (b)

(c)

(d)

(e)

Fig. 3. The manufacturing sequence for the Y-frame sandwich plates. (a) The leg is inserted into the slots in the folded Y-frame web. (b) The leg is laser welded to the Y-frameweb. (c) The Y-frame core is spot welded on to the front face sheet of the sandwich plate. (d) An aluminium insert locates the core and face sheets and (e) the assembly islaser welded together.

foam projectiles. The applied pressure versus time pulse mimicsshock loading in air and in water, with peak pressures on theorder of 100 MPa and loading times of approximately 100 μs.This laboratory method has been employed in several investiga-tions on the dynamic response of sandwich beams and plates,see for example Radford et al. (2006a) and Rathbun et al. (2006).Rubino et al. (2008b) have recently used this technique to demon-strate that sandwich beams with Y-frame and corrugated coreshave a superior dynamic performance to monolithic beams of equalareal mass; in these tests, the prismatic axis of the cores wasaligned with the longitudinal axis of the beams. However, clampedplates are more representative of marine structural components.The main aim of the current study is to determine the relativedynamic performance of clamped monolithic plates and clampedsandwich plates with Y-frame and corrugated cores. In order toenable a fair comparison with the beam results in Rubino et al.(2008b) we shall employ the same methodology to load the sand-wich plates investigated here.

The outline of the paper is as follows. First, the manufactur-ing route of the sandwich plates is detailed and the experimentalprotocol is described for dynamic loading the plates over a centralpatch by metal foam projectiles. The experimental results are thencompared with the corresponding observations for beams withtwo orientations of the core and also for monolithic plates of equalareal mass. Finally, finite element predictions are used to estimatethe onset of failure in the plates.

2. Experimental investigation

Clamped rectangular sandwich plates, made from AISI 304stainless steel and comprising two identical face-sheets and eithera Y-frame or corrugated core, were loaded dynamically with metalfoam projectiles as sketched in Fig. 2a. The main aims of this ex-perimental study were:


(a)

(b)

Fig. 4. The clamping arrangement of the sandwich plates. (a) The steel inserts and epoxy filling of the core ends allow for high clamping pressures. (b) The steel frame forclamping of the plates. All dimensions are in mm.

(i) To measure the shock resistance of rectangular sandwichplates with Y-frame and corrugated cores, and of monolithicplates. All plates have an equal areal mass.

(ii) To compare the dynamic performance of clamped rectangularsandwich plates and of end-clamped beams.

(iii) To investigate the fidelity of three-dimensional finite elementcalculations in predicting the observed dynamic responses.

2.1. Specimen configuration and manufacture

The core and face sheets of the sandwich plates were madefrom AISI type 304 stainless steel of thickness t = 0.3 mm andof density ρ f = 7900 kg m−3. The Y-frame and corrugated coreshad an effective density ρc ≈ 200 kg m−3 (corresponding to a rel-ative density ρ̄ = 2.5%) and a core depth c = 22 mm. All sandwichplates had an areal mass m = 2tρ f + cρc ≈ 10 kg m−2.

A co-ordinate system (X1, X2, X3) is introduced in Fig. 2a in or-der to clarify the orientations of the cores in the sandwich plates.The longitudinal and transverse axes of the Y-frame and corru-gated core are aligned with the X1 and X2 directions, respectively,while the through thickness direction of the core is along the X3axis. The short edge of the plate between the clamped edges is

labelled b along the X2 direction while the long edge is labelled2L along the X1 direction; the chosen aspect ratio b/2L = 0.52matches that of Royal Schelde’s hull panels. The total depth of thesandwich plates is denoted c + 2t . The cross-sectional dimensionsof the Y-frame and corrugated core are given in Figs. 2b and 2c,respectively. Note that the Y-frames and corrugated cores are close-packed along the X2-direction such that adjacent cores touch assketched in Fig. 2.

The manufacturing sequence for the Y-frame sandwich plates isoutlined in Fig. 3. The Y-frame webs were computer-numerical-control (CNC) folded to produce a continuous corrugated profilewith flat tops. Slots were then CNC-cut along the tops of the cor-rugations as seen in Fig. 3a. Each corrugated sheet contained fiveY-frame webs. The legs of the Y-frame were CNC folded and thenteeth were CNC-cut into the bottom of these legs to mate withthe slots of the corrugations (Fig. 3a). The assembly was thenlaser welded together (Fig. 3b). This core assembly was of length375 mm and of width 130 mm. Next, the Y-frame was spot weldedto 375 mm × 250 mm stainless steel face sheets (Figs. 3c and 3d).The face sheets were then continuous laser welded to the core us-ing CNC laser welding. In order to ensure intimate contact between


the face sheet and the core during the laser welding, an aluminiuminsert temporarily supported the core (Fig. 3d).

The corrugated core, comprising ten struts (five corrugations) in-clined at ±60◦ (Fig. 2c), was also manufactured by CNC folding.The core was then laser welded to two identical face sheets us-ing the same procedure as that described above for the Y-framesandwich plates.

In order to minimise rotations and deflections of the sandwichplates along the clamped periphery, the clamping procedure assketched in Fig. 4 was employed. Along the sides of the sandwichpanel, two steel inserts of dimension 375 mm × 60 mm × 22 mmwere glued between the face sheets. The ends of the sandwichcore were then dipped in epoxy resin to a depth of 62.5 mm,as sketched in Fig. 4a. The periphery of the sandwich plateswas then clamped between two steel frames using M8 bolts, asshown in Fig. 4b. The inner dimensions of the clamping framewere 250 mm × 130 mm and thus the effective dimensions ofthe Y-frame and corrugated core sandwich plates (i.e. spans ofthe plate between the clamped edges) were 2L = 250 mm andb = 130 mm.

Monolithic rectangular plates of thickness h = 1.2 mm werealso made from the AISI 304 stainless steel; they possessed thesame areal mass as the sandwich plates, m ≈ 10 kg m−2, and wereof overall dimension 375 mm × 250 mm. As for the sandwich case,the monolithic plates were clamped between the steel frame ofFig. 4b, giving an effective span between the clamped edges of2L = 250 mm and b = 130 mm.

2.2. Uniaxial response of the constituent materials

Tensile tests were performed on dog-bone specimens machinedfrom the as-received AISI 304 stainless steel material.2 The truestress versus logarithmic strain curve σ0(ε

p) of the AISI 304 stain-less steel measured at an applied nominal strain-rate of 10−3 s−1

is reported in Fig. 5a (note that the measured uniaxial stress ver-sus strain responses of the 0.3 mm and 1.2 mm steel sheets wereindistinguishable). The response is linear elastic with a Young’smodulus E = 210 GPa, followed by linear hardening above a yieldstrength of σY = 210 MPa. The finite element model requires adescription of the high strain rate response of the 304 stainlesssteel. The strain-rate sensitivity of the AISI 304 stainless steelwas documented by Stout and Follansbee (1986) in the range10−4 s−1 < ε̇ < 104 s−1. Following their approach, we define a dy-namic strength enhancement ratio R as the ratio of the strengthσd (at a plastic strain εp = 0.1) at an applied strain-rate ε̇p to thestrength σ0 (εp = 0.1) at an applied ε̇p = 10−3 s−1. Stout and Fol-lansbee (1986) found that the ratio R is reasonably independenton the choice of plastic strain εp at which R is calculated. Wemake use their measured values for R(ε̇p) and write the dynamicstrength σd versus plastic strain εp response in the decoupledform:

σd = R(ε̇p)σ0(εp). (1)

This prescription for the strain rate sensitivity of the stainless steelis used in the finite element calculations described in Section 4;σ0(ε

p) is the measured quasi-static stress versus strain history(Fig. 5a). For illustration, the estimated true tensile stress versuslogarithmic plastic strain behaviour of the AISI 304 stainless steelis sketched in Fig. 5a at four selected values of applied strain-rateusing the R values given by Stout and Follansbee (1986).

2 We anticipate that the welding process results in a heat affected zone whereinthe properties of the steel are affected. We neglect the influence of this heat af-fected zone and subsequently show that this assumption gives reasonable agree-ment between the finite element predictions and measurements.

(a)

(b)

Fig. 5. (a) The measured quasi-static (ε̇p = 10−3 s−1) tensile stress versus strainresponse of the AISI 304 stainless steel and the estimated responses at three addi-tional values of applied plastic strain rate, using the data of Stout and Follansbee(1986). (b) The quasi-static nominal compressive stress versus nominal strain re-sponse of the Alporas metal foam.

The projectiles used to impact the plates are made from Alporasaluminium alloy foam of relative density ρ̄ ≈ 0.11. The compres-sive response of the foam is required in order to calibrate the con-stitutive parameters of the finite element model. The quasi-staticcompressive response of the foam was measured at a strain-rateof 10−3s−1 using a cylindrical specimen similar to that of the pro-jectile in the dynamic experiments (diameter 50.8 mm and length51 mm). The nominal compressive response is shown in Fig. 5b:the foam has a plateau strength of approximately 2.5 MPa and anominal densification strain of εD ≈ 0.8.

2.3. Protocol for the dynamic tests

The clamped rectangular monolithic and sandwich plates wereimpacted by Alporas aluminium foam projectiles over a centralcircular patch of diameter d, as shown in Fig. 2a. Radford et al.(2005) have recently demonstrated that foam projectiles can beused to provide well-characterised pressure versus time loadingpulses. This method has been used to measure the dynamic re-sponse of sandwich beams with lattice cores (Radford et al., 2006a)and circular sandwich plates with metal foam cores (Radford etal., 2006b) and lattice cores (McShane et al., 2006). The impactsite of the Y-frame and corrugated core sandwich plates is markedin Fig. 2: in keeping with the Y-frame structures used in shipsconstructed by Schelde Shipbuilding, the web of the Y-frame was


Table 1The measured dynamic responses of monolithic and sandwich plates. The symbol Xdenotes through-thickness tearing of the plates

Specimen Projectile Response

I0 = ρpl0 v0

(kN s m−2)ρp

(kg m−3)v0

(m s−1)wb

(mm)εc Visual signs

of failure

Monolithic 1.6 331 94 10.5 – –3.0 306 191 15 – –5.1 334 299 23 – –5.5 308 351 26 – –6.2 311 392 28.8 – –6.6 289 451 33 – –

Y-frame 1.7 331 99 2 0.55 –3.1 306 198 6 0.84 longitudinal tear

on impacted face5.1 332 301 18 0.96 longitudinal tear

on impacted face5.4 300 351 X X failure of both face

sheets (projectilepenetrates)

Corrugated 1.7 331 99 2.5 0.49 –3.0 309 192 6 0.73 –5.0 332 297 16.5 0.96 longitudinal tear

on impacted face5.4 300 351 X X failure of both face

sheets (projectilepenetrates)

adjacent to the impacted face of the plate while the Y-frame legwas attached to the non-impacted face of the sandwich plate.

Projectiles of length l0 ≈ 51 mm and diameter d = 50.8 mmwere electro-discharge machined into a circular cylindrical shapefrom Alporas foam blocks of density ρp = 313 kg m−3. The projec-tiles were fired at a velocity v0 in the range 94 m s−1 to 451 m s−1

using a gas gun with circular barrel of diameter 50.8 mm andlength 4.5 m. Results are given in terms of the projectile momen-tum per unit area I0 = ρpl0 v0, as listed in Table 1. The plateswere inspected after each experiment in order to measure thepermanent central deflection and to detect any visible signs of fail-ure.

3. Experimental results

At least four levels of momentum I0 were imparted to themonolithic and sandwich plates by varying the metal foam pro-jectile impact velocity. A summary of the measured deflections,core compression and observed failure modes is provided in Ta-ble 1.

3.1. Effect of projectile momentum upon the deformation of monolithicand sandwich plates

The dynamic response of the sandwich and monolithic platesis summarised in a plot of permanent back face deflections wb atthe centre of the plates versus the initial momentum of the foamprojectile I0, see Fig. 6a. Both types of sandwich plate have a simi-lar performance over the projectile momentum range 2 kN s m−2 �I0 � 5 kN s m−2 and are perforated by the projectile at higher val-ues of I0. In contrast, the monolithic plates have higher deflectionswb than the sandwich plates but remain intact over the entirerange of I0 investigated.

Define the permanent core compressive strain as εc ≡ �c/c,where �c is the reduction in core thickness at the centre of theplate due to impact. The measured dependence of εc upon I0 isplotted in Fig. 6b for the two sandwich plate configurations. Bothsandwich plates undergo similar levels of core compression overthe range of I0 values investigated.

Photographs showing the impacted face of the as-tested plates(I0 ≈ 5 kN s m−2) within the clamping frame are shown in Fig. 7.

(a)

(b)

Fig. 6. Measured and FE predictions of the permanent (a) back face deflections wb

and (b) core compression at the centre of the dynamically loaded sandwich plates,as a function of the foam projectile momentum I0. The data for the monolithicplates are included. Symbols denote measurements while the continuous lines arethe FE predictions. Front face sheet failure is denoted by encircling the salient datapoints of in (b).

Tears running parallel to the X1-axis are observed on the im-pacted face sheets of both the Y-frame and corrugated core sand-wich plates. In contrast, the monolithic plate remained intact atthis value of I0. Partial pull-out of the monolithic plate from theclamped supports is evident in Fig. 7a: a pen-line was traced alongthe inner clamped boundaries of the plates prior to each test, andthe line moved inwards during the test.

In order to gain insight into the dynamic deformation mecha-nisms, the monolithic and sandwich plates tested at I0 ≈ 3 kN s m−2

were sectioned along their mid-span X1 = 0. Photographs of thesectioned specimens are shown in Fig. 8a for the monolithic plate,in Fig. 8b for the sandwich plate with the Y-frame core and inFig. 8c for the sandwich plate with the corrugated cores. The sec-tioned profiles show that significant plastic deformation occurs inthe vicinity of the foam impact site and that the plates are con-tinuously curved. It is deduced that the dynamic deformation ofplates involves the formation of travelling hinges, similar to theobserved behaviour of monolithic and sandwich beams and platesreported by numerous investigators, for example Radford et al.(2006b); McShane et al. (2006) and Rubino et al. (2008b). For bothtypes of sandwich plate, the core has sheared and compressed.


(a)

(b)

(c)

Fig. 7. Photographs showing a view of the front face of the as-tested (a) monolithic(b) Y-frame and (c) corrugated core plates. I0 ≈ 5 kN s m−2.

3.2. Comparison of the performance of rectangular sandwich plates andbeams

Rubino et al. (2008b) investigated the response of clamped Y-frame and corrugated core sandwich beams impacted by metalfoam projectiles. Beams with both transverse and longitudinal coreorientations were investigated in that study and constituent mate-rials used to manufacture the sandwich beams, the core and facesheet thicknesses and core relative density were identical to thoseemployed in the present investigation. Moreover, the spans of thebeams in Rubino et al. (2008b) were 250 mm, that is equal to thelength of the longer side of the rectangular plates studied here.Given these similarities between the two studies it is instructive tocompare the responses of the Y-frame and corrugated core sand-wich beams and plates. In qualitative terms, the deformed shapeon the X1 = 0 plane of the plate specimens (Fig. 8a) is in good

(a)

(b)

(c)

Fig. 8. Photographed sections of the mid-plane X1 = 0 of the as-tested (a) mono-lithic (b) Y-frame and (c) corrugated core plates. I0 ≈ 3 kN s m−2.

agreement with that observed by Rubino et al. (2008b) for corru-gated and Y-frame sandwich beams in the transverse configuration.

The permanent back face deflections wb at mid-spans of theY-frame and corrugated core sandwich beams and plates are com-pared in Figs. 9a and 9b, respectively. Recall that the sandwichplates have the longitudinal axis of the prismatic cores along theX1 axis while the transverse core direction is aligned along the X2

axis. However, the measured sandwich plate deflections are not in-termediate to the sandwich beams with transverse and longitudi-nal core arrangements. Rather, the sandwich plates outperform thesandwich beams (in terms of wb) with the longitudinal core ori-entation over the entire range of projectile velocities investigatedhere. This is presumably due to the enhanced stretch resistanceof the face sheets in the clamped plate configuration. However,the enhanced stretching of the face-sheets results in tearing ofthe front face sheet at lower values of I0 compared to the sand-wich beams with the longitudinal core orientation as indicatedin Fig. 9. Moreover, perforation of the sandwich plates occurs atI0 = 5.4 kN s m−2 which is about 15% lower than that for sand-wich beams with a longitudinal core orientation.

It is instructive to make a direct comparison of the deflectionsof the sandwich beams and plates with their monolithic coun-terparts. To address this, the ratio of measured permanent backface deflection at the centre of the sandwich plates to that ofmonolithic plates (of same mass) wsandwich/wmonolithic is plottedas a function of projectile momentum I0 in Fig. 10. The data forsandwich and monolithic beams from Rubino et al. (2008b) are in-cluded in the figure. There is only a minor effect of core topologyupon the mid-span deflection for the beams and plates. However,the sandwich plates and the beams with a longitudinal core orien-tation outperform the monolithic plates particularly at low impulselevels; in contrast, the sandwich beams with the transverse coreorientation show negligible benefit over monolithic beams of equalmass.

4. Finite element simulations

Finite element (FE) predictions for the monolithic and sand-wich plates are presented in this section. All computations wereperformed using the explicit time integration version of the com-


(a)

(b)

Fig. 9. Comparison of the measured permanent deflections wb at the centre of theback face of the (a) Y-frame and (b) corrugated core sandwich plates and beams, asa function of the foam projectile momentum I0. The data are shown up to I0 val-ues at which face sheet failure is first observed. The sandwich beam data are fromRubino et al. (2008b) and are for the cases when the prismatic axis and transverseaxis of the core are along the length of the beam; these orientations are labelledlongitudinal and transverse, respectively.

Fig. 10. The ratio of measured back deflections of the sandwich plates to monolithicplate versus projectile momentum I0. The data are shown up to I0 values at whichface sheet failure is first observed. The sandwich beam data are from Rubino et al.(2008b), with the prismatic and transverse axes of the core along the length of thebeam; these two cases are labelled longitudinal and transverse, respectively.

mercially available finite element code ABAQUS3 (version 6.5). Wefirst briefly describe the details of these FE calculations.

Three-dimensional simulations of half models of the rectangularsandwich and monolithic plates were performed. Linear hexahe-dral elements (C3D8R in ABAQUS notation) were used to model thecylindrical foam projectiles of diameter d = 50.8 mm and lengthl0 = 51 mm, with the elements swept about the cylindrical axisof the foam projectile. The projectiles had 51 elements in the ax-ial direction and 25 elements along the radius. The sandwich andmonolithic plates were modelled using four-nodded shell elements(S4R in ABAQUS notation) for both the face-sheets and the core.Perfect bonding between the core and face sheets was assumed.All dimensions (face sheet thickness and core dimensions) werechosen to match the experimental values and an element size of1.0 mm was employed for all calculations reported subsequently.

Symmetry boundary conditions were specified at mid-span(X1 = 0) and clamped boundary conditions were applied at X1 = Land at X2 = ±b/2 by constraining all rotations and displacementsto zero; see Fig. 2a. The loading was simulated by impact of ametal foam projectile: at the start of the simulation the projectilewith an initial velocity v0 was brought into contact with the frontface sheet at the centre of the rectangular plates. The “general con-tact” option in ABAQUS was employed to simulate contact betweenall adjacent surfaces. The general contact algorithm in ABAQUSenforced hard, no-friction contact interaction using a penalty al-gorithm.

4.1. Material properties

The AISI 304 stainless steel was treated as a J2-flow theoryrate dependent solid of density ρ f = 7900 kg m−3, Young’s mod-ulus E = 210 GPa and Poisson ratio ν = 0.3. The uniaxial tensiletrue stress versus equivalent plastic strain curves at plastic strain-rates 10−3 s−1 � ε̇p � 104 s−1 were tabulated in ABAQUS usingthe prescription described in Section 2.2 and employing the dataof Fig. 5a. Over the range of impact velocities considered here, adi-abatic heating effects are negligible and hence not included in theFE analysis.

The metal foam projectile was modelled as a compressiblecontinuum using the constitutive model of Deshpande and Fleck(2000). An isotropic yield surface is defined by:

σ̂ − Y = 0, (2)

where Y is the current yield strength. The equivalent stress σ̂ hasthe following form:

σ̂ 2 ≡ 1

1 + (α/3)2[σ 2

e + α2σ 2m], (3)

where the parameter α defines the shape of the yield surface,σm ≡ σkk/3 is the mean stress and σe ≡ √

3si j si j/2 is the vonMises effective stress in terms of the deviatoric stress si j . The yieldstrength Y is assumed to be given by an over-stress type relation

Y = μ ˙̂εp + σc, (4)

where μ is the foam viscosity, ˙̂εpis the work conjugate plastic

strain rate to σ̂ and σc(ε̂p) is the uniaxial stress versus plastic

strain relation at a negligible strain rate. In the simulations, theAlporas foam is ascribed a density of ρp = 313 kg m−3, Young’smodulus Ec = 1.0 GPa, an elastic Poisson’s ratio ν = 0.3. Consistentwith observations reported by Deshpande and Fleck (2000) we as-sumed α = 3/

√2 which implies that the “plastic Poisson’s ratio”

νp = 0. The yield strength σc versus equivalent plastic strain ε̂p

3 Hibbit, Karlsson and Sorensen Inc.


Fig. 11. Finite element prediction of the temporal variation of the deflection of thecentre of the monolithic and sandwich plates loaded with a metal projectile mo-mentum of I0 = 3.2 kN s m−2.

history is calibrated using the compressive stress versus strain re-sponse presented in Fig. 5b. The assumed over-stress model smearsthe plastic shock wave in the foam projectile over a width (Radfordet al., 2005):

w = μεD

ρp�v, (5)

where εD is the nominal densification strain and �v is the ve-locity jump across the shock. For the purposes of this discussion,�v is approximately equal to the projectile velocity v0. The val-ues of the viscosity μ were chosen so that w = l0/10 ≈ 5 mm,which is approximately equal to that observed in the experimentsof Radford et al. (2005). Large gradients in stress and strain occurover the shock width and thus the foam projectile was discretizedby a mesh size of 1 mm in order to resolve these gradients accu-rately.

Typically, in these experiments, the foam projectile fracture af-ter full densification. However, this fracture does not significantlyaffect the response of the plates as the foams fracture after theyhave transferred most of their momentum to the plates. We thusneglect foam fracture in the FE calculations.

4.2. Comparison of finite element predictions and measurements

Representative FE calculations of the transient deflection of thecentre point of the sandwich and monolithic plates are plotted inFig. 11 for I0 = 3.2 kN s m−2. The deflections of both the front andback face sheets of the sandwich plates are displayed. In all casesonly small elastic vibrations occur after the maximum deflectionshave been attained, so that the peak deflection is close to the per-manent deflection.

The predicted permanent deflections wb of the back face sheetsof the sandwich plates and of the monolithic plates are included inFig. 6a along with the corresponding measured values. The perma-nent deflections in the FE calculations are estimated by averagingthe displacements over several cycles of elastic vibration (fromtrough to peak) immediately after the initial peak displacement.We conclude that at least for the lower values of I0 where notearing of the plates was observed, the FE simulations are able topredict the plate deflections with reasonable accuracy. Some of thediscrepancies may be accounted for as follows:

(i) At the high values of I0 the observed deflections of the mono-lithic plates are greater than the predictions as the plates startto pull out of the supports, see Fig. 7a.

(ii) The FE calculations overestimate wb for the Y-frame plates atintermediate values of I0. We shall see subsequently that thisis primarily due to the fact that the FE calculations underesti-mate the degree of core compression in the Y-frame plates.

The FE predictions for the final core compression εc at the platecentre are compared with measurements in Fig. 6b as a func-tion of the applied momentum I0. Good agreement between themeasurements and predictions is achieved for the corrugated core,however the FE calculations under-predict the core compressionin the Y-frame sandwich plates. The critical difference betweenthe predictions and observations is that the FE calculations pre-dict that the core compression remains constant at εc ≈ 0.4 forI0 � 3 kN s m−2 while in the experiments core compression in-creases with increasing I0 over the entire range of projectile ve-locities investigated here. We attribute this discrepancy to the factthat tearing between the Y-frame leg and the back face sheet wasobserved in all specimens but was not modelled in the FE cal-culations. It is also worth noting that the slight kink in the FEpredictions of the deflections of the back face of the Y-frame plates(Fig. 6a) corresponds to the I0 range over which the FE calculationspredict that the core compression remains constant.

Comparisons between the measured and predicted permanentdeformation of the plates impacted at I0 ≈ 3 kN s m−2 are includedin Fig. 8. In both the experiments and the FE calculations the platesare sectioned along their mid-span X1 = 0 to reveal the core de-formation in the sandwich plates. In general good agreement isobtained although there are some discrepancies especially for theY-frame sandwich plate as discussed above.

It is worth emphasising here that all tests and simulations inthis study were carried out on approximately 1/20 scale speci-mens made from 304 stainless steel. Finite element calculationsof Tilbrook et al. (2007) suggest that at-least the dynamic com-pressive response of these laboratory scale Y-frame and corrugatedcore specimens is representative of the full-scale structures. Ad-ditional finite element calculations and larger scale experimentsare required to understand whether this finding extends to scaled-down clamped plates as well.

4.3. Estimation of the onset of front face tearing

The final plastic strain distribution within the Y-frame and cor-rugated core sandwich plates was explored in the FE simulationsfor the choice I0 = 5.1 kN s m−2. This impulse level was chosen asit gave front face tearing of both sandwich plates, but not completeperforation. The location and direction of the maximum principalstrains in the sandwich plates are shown in Figs. 12a and 12bfor the Y-frame and corrugated core plates, respectively. For theY-frame plate, the highest predicted strain occurs close to the jointbetween the Y-frame web and the front face sheet while for thecorrugated core high values of final plastic strain occur both in thefront face sheet and within the core itself. The plastic strain pre-dictions of the FE calculations are generally consistent with theobservation that failure always initiates in the front face sheet.

The FE predictions of the maximum principal plastic strainsεmax in the face sheets of the Y-frame and corrugated core platesare compared with those in the corresponding sandwich beams(longitudinal core orientation) in Fig. 13a over a range of valuesof I0. The FE predictions for the beams are obtained from the par-allel study by Rubino et al. (2008b). The FE calculations predictsignificantly higher values of εmax in the plates than in the beams,especially for I0 � 4 kN s m−2; this is consistent with the observa-tion that the plates fail at lower values of I0 than the sandwich


(a)

(b)

Fig. 12. Sketch representing the FE predictions for the location and direction ofthe maximum principal strain in the (a) Y-frame and (b) corrugated core sandwichplates. The beams were impacted by the foam projectile of I0 = 5.1 kN s m−2.

beams of longitudinal core orientation. Comparisons between thepredicted values of εmax for the sandwich and monolithic platesare included Fig. 13b. Recall that failure of the front face sheetwas observed at lower values of I0 in the Y-frame plate than thecorrugated core plate. Also, no failure of the monolithic plate wasobserved even at the highest value of I0 investigated here. Consis-tent with these observations, the FE calculations predict the lowestvalues of εmax in the monolithic plates followed by the corrugatedcore plates, and the largest strains occur in the Y-frame plates.

5. Concluding remarks

Metal foam projectiles have been used to impact clamped 304stainless steel monolithic plates and sandwich plates with Y-frameor corrugated cores. The permanent deflections and level of corecompression of the sandwich plates have been measured as a func-tion of projectile momentum, and the measured responses arecompared with finite element simulations. The finite element (FE)simulations capture the observed deformation modes to reason-able accuracy. Failure of the joints between the core and the facesheets is not included in the FE predictions, and this leads to somediscrepancies especially for the Y-frame plates.

The sandwich plates outperform monolithic plates of equalmass at low values of the projectile momentum I0. Moreover,comparisons with the corresponding Y-frame and corrugated coresandwich beams revealed that the plates deflect less than thebeams. However, the benefits of sandwich construction reducewith increasing I0: (i) the ratio of the permanent deflections of thesandwich plates to monolithic plates increases with increasing I0and approaches unity for I0 ≈ 6 kN s m−2; (ii) stress concentrationsat the attachment points between the core and the face sheets re-sult in failure of the sandwich plates at lower values of I0 thanfor the monolithic plates. We conclude that sandwich constructionhas potential for significantly enhancing the dynamic performanceof structures. However, sandwich construction must be used withcaution in heavily loaded structures: the sandwich plates may failat lower values of I0 than for monolithic plates of equal areal mass.

(a)

(b)

Fig. 13. FE predictions of the maximum principal plastic strain in the front facesheet of the dynamically loaded sandwich plates. (a) Comparison of the Y-frameand corrugated core sandwich beams (longitudinal core orientation) and plates.(b) Comparison between the sandwich plates and monolithic plates. The FE pre-dictions for the sandwich beams are taken from Rubino et al. (2008b).

Acknowledgements

This work was supported by the Netherlands Institute for MetalResearch: project number MC1.03163, The Optimal Design of Y-coresandwich structures.

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